On the Numerical Solution of Nonlinear Eigenvalue Problems for the Monge-Amp\`{e}re Operator

TL;DR

This paper develops a numerical method for solving nonlinear eigenvalue problems involving the Monge-Ampère operator, using divergence formulation, operator-splitting time discretization, and finite element approximation, demonstrating convergence in tested cases.

Contribution

It introduces a novel numerical approach combining divergence formulation, operator-splitting, and finite elements for Monge-Ampère eigenvalue problems, validated through convergence tests.

Findings

Method converges to known solutions as discretization refines

Applicable to problems without known exact solutions

Demonstrates effectiveness of the combined numerical approach

Abstract

In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Amp\`{e}re operator $v\rightarrow \det \mathbf{D}^2 v$. The methodology we employ relies on the following ingredients: (i) A divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by operator-splitting of an initial value problem (a kind of gradient flow) associated with each eigenvalue problem. (iii) A finite element approximation relying on spaces of continuous piecewise affine functions. To validate the above methodology, we applied it to the solution of problems with known exact solutions: The results we obtained suggest convergence to the exact solution when the space discretization step $h\rightarrow 0$. We considered also test problems with no known exact solutions.